/ Technics/ Report No. 32-931 Large Deflection and Stability Analysis by the Direct Stiffness Method ffarold C. Martin h. GPO PRICE $ CFSTI PRICE(S) $ (THRU) / (CODE) ! Microfiche (MF) ~,f2 f! 653 July 65 ET PROPULSION LABORATORY CALIFORNIA lNSTlTUTE OF TECHNOLOGY PASAD EN A, CALIF 0 R N IA August 1, 1966 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Technical Report No. 32-931 Large Deflection and Stability Analysis by the Direct Stiffness Method Harold C. Martin 2.&. v M. E. Alper, Manager Applied Mechanics Section JET PROPULSION LABORATORY CALIFORNIAINSTITUTE OF TECHNOLOGY PASADENA,CALIFORNIA August 1, 1966 \ Copyright @ 1966 Jet Propulsion Laboratory California Institute of Technology Prepared Under Contract No. NAS 7-100 National Aeronautics &? Space Administration I I JPL TECHNICAL REPORT NO. 32-931 PREFACE This Report summarizes research conducted by the author as a consultant to the Jet Propulsion Laboratory. The author is Professor of Aeronautical and Astronautical Engineering at the University of Washington, Seattle, Washington. This work was originally prepared while the author was on leave of absence (1962-63) as Visiting Pro- fessor of Structural Mechanics in the Department of Civil Engineering, University of Hawaii, Honolulu, Hawaii. 111 I JPL REPORT NO 32-931 I TECHNICAL . CONTENTS .................... ! . 1. Introduction 1 II. Some Comments on the Nonlinear Theory of Eiasiicity .... -9 I . A. General Discussion .................. 2 B . Deformation of Volume Element ............. 2 C . Strain-Displacement Equations ............. 3 D. Physical Strains ................... 4 E . Rotations ..................... 4 F. Theory of Small Deformations .............. 5 G . Small Deformations and Small Angles of Rotation ...... 5 H . Reduction to Classical Theory .............. 6 I . Equilibrium Equations ................ 6 J . Concluding Comments ................ 7 111. The Axial Force Member ................ 8 A . Examples-Nonlinear Behavior .............. 8 B . Stiffness Matrix-Axial Force Member ........... 9 C . Application to Simple Problems ............. 12 D . Concluding Comments ................15 IV. The Piecewise linear Calculation Procedure ........ 15 A . Incremental Step Procedure-Discussion .......... 15 B. Simple Truss-Exact Soiution ..............16 C . Simple Truss-Incremental Step Solution ..........16 V . The Beam-Column ..................19 A . Present State of Development ..............19 B . Stiffness Matrix-Beam-Column .............19 C. Column Stability-Example ...............22 D . Tie Rod Deflection-Example ..............23 E.Comments. .................... 24 VI . Stability Calculations .................25 A . Preliminary Comments ................25 B. Initial Loading Intensity Factor ............. 25 C . Plotting the Determinant of the Stiffness Matrix ........25 D . Adaptation of Southwell's Method ............26 E . Stability of Rectangular Plate-Example ..........26 F. Matrix Iterative Procedure ...............28 G . Stability of Tapered Column ..............28 H . Additional Cases ..................29 V I JPL TECHNICAL REPORT NO. 32-931 7 CONTENTS (Cont’dl VII. The Thin Plate Element . 29 A. Introductory Remarks . , . 29 B. Membrane Stiffness Matrices . 29 .I C. Bending Stiffness Matrices . , . 29 D. Initial Force Stiffness Matrices . 30 Nomenclature . , . 31 References . 31 Table 1. Exact solution data-truss of Fig. 1 . 16 Table 2. Computer results-rectangular plate stability problem . 27 FIGURES 1. Extensible rod-spring system . .... a 2. load-displacement curves . .... a 3. Heated beam-column . .... 9 4. Beam-column . .... 9 5. Tension -compression member (stringer) . .... 10 6. Arbitrarily oriented tension-compression member . .... 12 7. Nonlinear truss . ...*12 8. Truss stability problem . ....14 9. Solutions for truss of Fig. 1 . , . ....16 10. Beam-column nodal forces and displacements . .... 19 1 11. loading for simple beam-column . ....23 12. Extrapolation for critical load . ....26 13. Application of Southwell’s method . ....26 14. Rectangular plate stability problem . ....26 15. Idealized plate quadrant containing 32 triangular elements ....27 16. Southwell type plot for plate stability problem . ....28 17. Tapered column . , . ....28 VI JPL TECHNICAL REPORT NO. 32-931 ABSTRACT The application of the direct stiffness method in the solution of large deflection and stability problems is demonstrated. Discussed first are the basic elasticity equations that contain higher order terms to account for the nonlinear character of large deflections and large rotations. Various simplifications of these equations are made, and the conditions and limitations of the resulting expressions are discussed. Next, examples of geometrically nonlinear systems are presented to illustrate the importance of considering the nonlinear behavior in many problems. Finally, the modifications to the stiffness matrix, and the method of formal solution are derived for some of the nonlinear examples that have been discussed. m 1. INTRODUCTION The basic purpose of this Report is to show how the structural stiffness matrix. This in turn will enable the direct stiffness method may be extended to apply to structural dynamicist to determine the natural frequencies geometrically nonlinear structural problems. Of particular and principal modes of the prestressed, and subsequently interest are problems involving the deformation of bodies loaded, structure. A case in point is a body subjected to having initial stresses (such as those due to heating) and nonuniform thermal gradients and then acted on by problems in structural stability. As will be seen subse- external static and inertia forces. In this instance the self- quently, the conventional stiffness procedure is incapable equilibrating, initial (thermal) stresses must be taken into of handling such problems. Not only must new s&ess account when calculating structural stiffness against the matrices be derived, but the method for using these must subsequently applied external loadings. also be established. It must be emphasized that the direct stiffness method A piecewise linear (incremental step) procedure will be is essentially a matrix numerical procedure intended to introduced for determining displacements in the large de- be camed out on high speed digital computing equip flection problem. Stability of complex structural systems ment. It has inherent characteristics which make it highly will be put into a mathematical form such that the widely suited to such implementation. By the same token, it is used matrix iterative procedure may be applied in deter- only suitable for the simplest problems when hand cal- mining critical loadings and corresponding mode shapes. culations are to be used. It is nevertheless very important that carefully chosen problems, sufficiently simple for hand The basic element stiffness matrices, plus the piecewise calculation, be devised to illustrate the calculation pro- linear usage of these, will yield the overall instantaneous cedure. Such caidatiuns are iiichided in thii Reprt. 1 JPL TECHNICAL REPORT NO. 32-931 I II. SOME COMMENTS ON THE NONLINEAR THEORY OF ELASTICITY I A. General Discussion 4. Problems which are both physically and geometri- cally nonlinear I This section of the Report presents a brief outline of the classical nonlinear theory of elasticity. It has been abstracted from Ref. 1. The first type represents the classical elasticity problem. It exists when angles of rotation are of the same order of magnitude as elongations and shears, and when the latter A thorough familiarity with the basic nonlinear theory I are small compared to unity. Also, strains do not exceed is not essential for reading subsequent sections of this the limit of proportionality. Report. Some of the concepts and resulting equations from I the nonlinear theory will prove to be useful, however. The third type is the one of interest here. Again, strains I are assumed to not exceed the proportional limit, but rota- I Nonlinearity is introduced into the theory of elasticity in , tions may be large. This means that rotations may not be three ways: neglected in either the straindisplacement equations, or 1. Through the strain-displacement equations in writing the equilibrium equations. A thin, flexible, steel strip bent into a full circle is an example of this type of 2. Through the equilibrium equations problem. Less obvious cases will be mentioned shortly. 3. Through the stress-strain equations The general problem types which can only be investi- In the stress-strain equations, the nonlinear terms gated by using the nonlinear (geometric) theory of elastic- I appear when the strains exceed the proportional limit of ity are as follows: the material. This condition is therefore termed physical 1. Stability of elastic equilibrium nonlinearity. It is not treated in this Report. 2. Deformation of bodies having initial stresses In the first two sets of equations listed above, the non- linear terms arise from geometrical considerations. The 3. Large deflection of rods fundamental factor is this: the angles of rotation must be taken into account in determining, (a) changes in length 4. Torsion and bending in presence of axial forces of line elements, and (b) formulating the conditions of equilibrium of the volume element. These are termed 5. Bending of plates and shells under deflections of the geometric nonlinearities and underlie the basic problems order of magnitude of the thickness to be treated in this Report. Physical and geometric nonlinearities are independent